Olympiad problems

2017 Kosovo Team Selection Test, 5

Tags: geometry

Given triangle $ABC$. Let $P$, $Q$, $R$, be the tangency points of inscribed circle of $\triangle ABC$ on sides $AB$, $BC$, $AC$ respectively. We take the reflection of these points with respect to midpoints of the sides they lie on, and denote them as $P',Q'$ and $R'$. Prove that $AP'$, $BQ'$, and $CR'$ are concurrent.

2006 AMC 8, 18

Tags: geometry , 3d geometry

A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white? $ \textbf{(A)}\ \dfrac{1}{9} \qquad \textbf{(B)}\ \dfrac{1}{4} \qquad \textbf{(C)}\ \dfrac{4}{9} \qquad \textbf{(D)}\ \dfrac{5}{9} \qquad \textbf{(E)}\ \dfrac{19}{27}$

2015 Argentina National Olympiad Level 2, 2

Tags: rectangle , geometry

Let $ABCD$ be a rectangle with sides $AB=3$ and $BC=2$. Let $P$ be a point on side $AB$ such that the bisector of $\angle CDP$ passes through the midpoint of $BC$. Calculate the length of segment $BP$.

1995 Baltic Way, 16

Tags: geometry

In the triangle $ABC$, let $\ell$ be the bisector of the external angle at $C$. The line through the midpoint $O$ of $AB$ parallel to $\ell$ meets $AC$ at $E$. Determine $|CE|$, if $|AC|=7$ and $|CB|=4$.

Novosibirsk Oral Geo Oly IX, 2019.5

Tags: geometry , fixed , tangent circle

Point $A$ is located in this circle of radius $1$. An arbitrary chord is drawn through it, and then a circle of radius $2$ is drawn through the ends of this chord. Prove that all such circles touch some fixed circle, not depending from the initial choice of the chord.

1990 Canada National Olympiad, 3

Tags: incenter , geometry , angle bisector , cyclic quadrilateral

The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral $q$. Show that the sum of the lengths of each pair of opposite sides of $q$ is equal.

Novosibirsk Oral Geo Oly VIII, 2021.3

Tags: geometry , angle

Find the angle $BCA$ in the quadrilateral of the figure. [img]https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png[/img]

2005 China Team Selection Test, 1

Tags: incenter , power of a point , geometry , geometric transformation , trigonometry , homothety

Triangle $ABC$ is inscribed in circle $\omega$. Circle $\gamma$ is tangent to $AB$ and $AC$ at points $P$ and $Q$ respectively. Also circle $\gamma$ is tangent to circle $\omega$ at point $S$. Let the intesection of $AS$ and $PQ$ be $T$. Prove that $\angle{BTP}=\angle{CTQ}$.

2012 JHMT, 2

Tags: geometry

A circle with radius $1$ has diameter $AB$. $C$ lies on this circle such that ratio of lengths of arcs $AC /BC= 4$. $\overline{AC}$ divides the circle into two parts, and we will label the smaller part Region I. Similarly, $\overline{BC}$ also divides the circle into two parts, and we will denote the smaller one as Region II. Find the positive difference between the areas of Regions I and II.

2023 ELMO Shortlist, G4

Tags: geometry

Let $D$ be a point on segment $PQ$. Let $\omega$ be a fixed circle passing through $D$, and let $A$ be a variable point on $\omega$. Let $X$ be the intersection of the tangent to the circumcircle of $\triangle ADP$ at $P$ and the tangent to the circumcircle of $\triangle ADQ$ at $Q$. Show that as $A$ varies, $X$ lies on a fixed line. [i]Proposed by Elliott Liu and Anthony Wang[/i]

2001 Romania Team Selection Test, 2

Tags: geometry , symmetry

The vertices $A,B,C$ and $D$ of a square lie outside a circle centred at $M$. Let $AA',BB',CC',DD'$ be tangents to the circle. Assume that the segments $AA',BB',CC',DD'$ are the consecutive sides of a quadrilateral $p$ in which a circle is inscribed. Prove that $p$ has an axis of symmetry.

2009 Belarus Team Selection Test, 1

Tags: equal segments , circles , tangent , geometry , tangent circle

Two equal circles $S_1$ and $S_2$ meet at two different points. The line $\ell$ intersects $S_1$ at points $A,C$ and $S_2$ at points $B,D$ respectively (the order on $\ell$: $A,B,C,D$) . Define circles $\Gamma_1$ and $\Gamma_2$ as follows: both $\Gamma_1$ and $\Gamma_2$ touch $S_1$ internally and $S_2$ externally, both $\Gamma_1$ and $\Gamma_2$ line $\ell$, $\Gamma_1$ and $\Gamma_2$ lie in the different halfplanes relatively to line $\ell$. Suppose that $\Gamma_1$ and $\Gamma_2$ touch each other. Prove that $AB=CD$. I. Voronovich

1995 Polish MO Finals, 3

Tags: geometry

$PA, PB, PC$ are three rays in space. Show that there is just one pair of points $B', C$' with $B'$ on the ray $PB$ and $C'$ on the ray $PC$ such that $PC' + B'C' = PA + AB'$ and $PB' + B'C' = PA + AC'$.

2011 Sharygin Geometry Olympiad, 5

Tags: equal segments , cyclic quadrilateral , geometry

A line passing through vertex $A$ of regular triangle $ABC$ doesn’t intersect segment $BC$. Points $M$ and $N$ lie on this line, and $AM = AN = AB$ (point $B$ lies inside angle $MAC$). Prove that the quadrilateral formed by lines $AB, AC, BN, CM$ is cyclic.

VI Soros Olympiad 1999 - 2000 (Russia), 9.5

Tags: bicentric quadrilateral , geometry

Given a circle $\omega$ and three different points $A, B, C$ on it. Using a compass and a ruler, construct a point $D$ lying on the circle $\omega$ such that a circle can be inscribed in the quadrilateral $ABCD$ (points $A$, $B$, $C$, $D$ must be located on circle $\omega$ in the indicated order).

2017 Balkan MO Shortlist, G5

Tags: projective geometry , concurrency , geometry , circumcircle

Let $ABC$ be an acute angled triangle with orthocenter $H$. centroid $G$ and circumcircle $\omega$. Let $D$ and $M$ respectively be the intersection of lines $AH$ and $AG$ with side $BC$. Rays $MH$ and $DG$ interect $ \omega$ again at $P$ and $Q$ respectively. Prove that $PD$ and $QM$ intersect on $\omega$.

1992 Poland - Second Round, 4

Tags: geometry , equal segments , circles

The circles $k_1$, $k_2$, $k_3$ are externally tangent: $k_1$ to $k_2$ at point $A$, $k_2$ to $k_3$ at point $B$, $k_3$ to $k_4$ at point $C$, $k_4$ to $k_1$ at point $D$. The lines $AB$ and $CD$ intersect at the point $S$. A line $ p $ is drawn through point $ S $, tangent to $ k_4 $ at point $ F $. Prove that $ |SE|=|SF| $.

2016 BMT Spring, 16

Tags: analytic geometry , tetrahedron , 3d geometry , geometric inequality , geometry , sphere

What is the radius of the largest sphere that fits inside the tetrahedron whose vertices are the points $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$?

1976 IMO Longlists, 39

Tags: geometry , geometric transformation , homothety

In $ ABC$, the inscribed circle is tangent to side $BC$ at$ X$. Segment $ AX$ is drawn. Prove that the line joining the midpoint of $ AX$ to the midpoint of side $ BC$ passes through center $ I$ of the inscribed circle.

2010 Iran Team Selection Test, 8

Tags: trapezoid , circumcircle , geometry

Let $ABC$ an isosceles triangle and $BC>AB=AC$. $D,M$ are respectively midpoints of $BC, AB$. $X$ is a point such that $BX\perp AC$ and $XD||AB$. $BX$ and $AD$ meet at $H$. If $P$ is intersection point of $DX$ and circumcircle of $AHX$ (other than $X$), prove that tangent from $A$ to circumcircle of triangle $AMP$ is parallel to $BC$.

Kyiv City MO 1984-93 - geometry, 1985.9.5

Tags: parallelogram , geometry , equilateral

Outside the parallelogram $ABCD$ on its sides $AB$ and $BC$ are constructed equilateral triangles $ABK$, and $BCM$. Prove that the triangle $KMD$ is equilateral.

2014 Contests, 3

Tags: tetrahedron , combinatorics , geometry , 3d geometry , sphere

A real number $f(X)\neq 0$ is assigned to each point $X$ in the space. It is known that for any tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have : \[ f(O)=f(A)f(B)f(C)f(D). \] Prove that $f(X)=1$ for all points $X$. [i]Proposed by Aleksandar Ivanov[/i]

III Soros Olympiad 1996 - 97 (Russia), 11.4

Tags: circles , geometry , tangent circle

There are four circles. The chord$ AB$ is drawn in the first one, and the distance from the midpoint of the smaller of the two formed arcs to $AB$ is equal to $1$. The second, third and fourth circles are located inside the larger segment and touch the chord $AB$. The second and fourth circles touch internally the first and externally the third. The sum of the radii of the last three circles is equal to the radius of the first circle. Find the radius of the third circle if it is known that the line passing through the centers of the first and third circles is not parallel to the line passing through the centers of the other two circles.

2010 Sharygin Geometry Olympiad, 2

Tags: circumcircle , geometry , vertices

Two intersecting triangles are given. Prove that at least one of their vertices lies inside the circumcircle of the other triangle. (Here, the triangle is considered the part of the plane bounded by a closed three-part broken line, a point lying on a circle is considered to be lying inside it.)

2019 Nigerian Senior MO Round 4, 2

Tags: midpoint , geometry , perpendicular , circles

Let $K,L, M$ be the midpoints of $BC,CA,AB$ repectively on a given triangle $ABC$. Let $\Gamma$ be a circle passing through $B$ and tangent to the circumcircle of $KLM$, say at $X$. Suppose that $LX$ and $BC$ meet at $\Gamma$ . Show that $CX$ is perpendicular to $AB$.